25 problems

Let $B$ be a ball, let $p>1$, and let $u\in W^{1,r}(B)$ with $r>p-1$. Suppose that $u$ solves … in the distributional sense. Iwaniec-Sbordone conjecture. Then…

Borisov's conjecture. The Nash–Kuiper theorem should extend to $\C^{1,\alpha}$-regularity whenever

Critical exponent conjecture. The critical value of $b1$ for the rigidity-flexibility dichotomy is one of

Convex integration relaxation conjecture. The obtained relaxation would yield the matrix inequality

Let $k\geq 3$ and consider the $k$-generalized Korteweg-de Vries equation on the periodic domain, with initial data in the Sobolev space $H^s(\mathbb T)$. Unconditional well-posedn…

Entropy-production conjecture. The conclusion of Theorem $$ should hold for every

Further-inequality conjecture. Some further $\Lambda$-convex inequality should be added to $U$ so as to reduce it appropriately and exclude the case

The Hölder exponent $\alpha$ measures the regularity of a $C^{1,\alpha}$ isometric embedding, and the preceding flexibility and rigidity results leave the precise threshold between…

Buckmaster–Vicol magnetic helicity conservation conjecture. The space $L^3_{t,x}$ is the threshold regularity space for solutions of the ideal MHD system to conserve magnetic helic…

Wild non-uniqueness conjecture. For any such Riemann initial data, there exists a smooth pressure law $p$ with $p'>0$ such that the corresponding system admits infinitely many boun…

Higher-dimensional decomposition conjecture. For every $D\in\mathcal{C}^{m,\gamma}(\bar\omega,\mathbb{R}^{d\times d}_{\mathrm{sym}})$,

Let $B$ be the magnetic field of an ideal magnetohydrodynamic solution, and let $L^3_{t,x}$ denote the space of space-time functions integrable to the third power. Faraco–Lindberg–…

Cheskidov–Luo's conjecture. Non-uniqueness of weak solutions should hold throughout the full supercritical regime

Non-uniqueness conjecture. Under the stated integrability condition and suitable assumptions on $\beta$ and $T$, such a density $\rho$ and incompressible Sobolev vector field $u$ e…

Let $M$ be a $6$-dimensional manifold, and let $F\subset TM$ be a rank-$4$ distribution of elliptic type that is maximally non-involutive. Elliptic (4,6) h-principle conjecture. Th…

Let $M$ be a $6$-dimensional manifold, and let $F\subset TM$ be a rank-$4$ distribution of elliptic type that is maximally non-involutive. Elliptic (4,6) h-principle conjecture. Th…

Convex-combination representation conjecture. The convex integration scheme should be applicable so that some convex combination of two such weak solutions admits the representatio…

Isett–Oh conjecture. For such weak solutions, the time-regularity should generically be minimal.

Isett–Oh conjecture. For every $\theta<\frac{1}{3}$, there exists a weak Euler solution $v\in C^\theta(\mathbb R\times\mathbb T^n)$ such that

Modena–Székelyhidi threshold conjecture. The threshold in the existing convex-integration constructions is not optimal; rather, the optimal threshold should be

Let $\Sigma$ be a $2$-dimensional compact Riemannian manifold, possibly with boundary, with positive Gauss curvature. A $C^{1,\alpha}$ isometric embedding is an isometric embedding…

Let $\Sigma_a$ be the polar cap and let $\mathscr{I}^\alpha_1(\Sigma_a)$ denote the class of $C^{1,\alpha}$ isometric embeddings of $\Sigma_a$ into the relevant codimension-one Euc…

Consider admissible weak solutions of the compressible barotropic Euler system, and compare their energy dissipation over time. Maximal-dissipation conjecture. The principle of max…

Let $m$ and $q$ be positive integers, and let ${\cal D}_{m,q}$ denote the operator … from $C^\infty(\mathbb R^m,\mathbb R^q)$ to the space of Riemannian metrics on $\mathbb R^m$, w…

Let the BI manifold be the $6$-dimensional submanifold of $(h,Q,D,B)\in R^{10}$ defined by … The augmented Born–Infeld (ABI) system is the $10\times 10$ system obtained by adjoinin…